Non-uniform motion

Following the previous post on uniform motion, this post covers non-uniform motion in space and time.

A simple way to get non-uniform motion is to join two uniform motions in different directions; the change in direction means the joint motion is non-uniform. Another way to get non-uniform motion is to accelerate at a constant rate. Another way is to change from one uniform rate to another uniform rate.

The point of this post is that whereas space and time were interchangeable for uniform motion, for non-uniform motion space and time are not interchangeable. Spatial and temporal measures of non-uniform motion are not proportionate. Non-uniform motion requires that space and time be considered separately, and this includes all components and directions.

I previously gave an example of two uniform motions in different directions, here. What follows is an example of two uniform circular motions.

Consider an old-fashioned turntable for records (aka vinyl). Say a mark on it moves at 33 1/3 rpm (5/9 rps) for two and a half revolutions. Then the mark moves five and a quarter revolutions at 45 rpm (3/4 rps). The time taken is thus (9/5)(5/2) = 4.5 sec plus (4/3)(21/4) = 7.0 sec, which totals 11.5 sec. The final spatial angle is one half of a revolution or 180°. The temporal angle is (7.0/60)(360) = 42°, with the second hand of a circular clock as the standard circular motion.

Another standard of circular motion is the daily cycle of the sun’s position relative to the earth. This is divided into 24 hours of 15 degrees each. Say someone turns a post in the ground for 20 minutes at a rate of four revolution per hour. Then they get help and turn the post for five minutes at a rate of 10 revolutions per hour. What are the spatial and temporal angles of the post at the end? The spatial angle is (4/60)(20) = (4/3) rev plus (10/60)(5) = (5/6), which totals 13/6 of a revolution. The sun will have taken 20+5 = 25 minutes, which is (25/60)(15) = 6.25 degrees.

The temporal angle is the angle of a synchronous circular motion at the standard rate.